As a follow-up, would English (language) in his lifetime have included "lack" and "land" or are these words modernizations of an older form of English?

This is the part of the question that I'll be addressing here. Someone else will have to address the earliest source of the name.

According to Kate Northgate, a British historian from a century ago who wrote a biography on King John, the name was given to him by his father Hentry II around the time of his birth. The English "lackland" is a translation of the Norman French Johan Sanz Terre, sanz terre meaning "without land".

From Northgate's John Lackland:

It was in fact Henry himself who gave to his new-born child the name which has clung to him ever since—"Johans Sanz Terre", John Lackland

You can read Northgate's account here if you're interested. Not exactly a primary source, though I'm not sure you're going to find an account from his lifetime unless you've got some wicked sweet museum/library access.

Now, whether that story is itself true or not, the likelihood that the original phrasing would have been in Norman French rather than Early Middle English is significant. Assuming the tale of the naming is true, then it would have likely come into modern English by way of translation from the Norman French.

Then, as for whether the English language at the time had the words "lack" and "land": Old English definitely had "land" which has survived into modern English relatively untouched. I'm not sure "lack" is attested in English until after King John's death, however it did exist at the time in the Dutch and Frisian languages spoken at the time, coming from Proto-Germanic *laka. It might have existed in English as well, or been borrowed from a neighbouring language, or it might have only been applied later as a translation from the Norman. We simply don't have any attested records of it in Early Middle English before King John's lifetime that I'm aware of. That doesn't mean they don't exist buried somewhere that a scholar of Early Middle English might have access to.

edit: /u/rusoved has kindly pointed out that I'm using linguist-speak and that as a result my intended meaning on a certain point is probably not clear to the general reader of this subreddit. When I say it's attested or not, I'm not making the claim that if it isn't attested it doesn't exist. Instead I'm simply saying that I'm not comfortable saying "yes it existed" without having a written record of its use. We can absolutely assume that it probably did exist, since we know it was in the ancestor language of Middle English, and it's present in related language of the time period. My personal belief is that it surely must have existed, but lacking a specific use in a written record, I don't want to say that, because then it's speculative on my part.